Variational Representations related to Quantum Rényi Relative Entropies

In this note, we focus on the variational representations of some matrix norm functions and matrix trace functions that are related to the quantum R\'{e}nyi relative entropies. Concretely, by using the H\"{o}lder inequality and Young inequality for symmetric norms we give the variational representations of the function $(A,B)\mapsto \|B^{q/2}K^*A^pKB^{q/2}\|$ for symmetric norms. These variational expressions enable us to give some new proofs of the convexity/concavity of the trace function $(A,B)\mapsto Tr (B^{q/2}K^*A^pKB^{q/2})$ and some extensions of the Lieb's theorems in terms of symmetric norms or symmetric anti-norms.